arXiv:1312.2476v1 [math.AP] 29 Nov 2013
P-ADIC ELLIPTIC QUADRATIC FORMS, PARABOLIC-TYPE
PSEUDODIFFERENTIAL EQUATIONS WITH VARIABLE
COEFFICIENTS, AND MARKOV PROCESSES.
O. F. CASAS-SÁNCHEZ AND W. A. ZÚÑIGA-GALINDO
Abstract. In this article we study the Cauchy problem for a new class of
parabolic-type pseudodifferential equations with variable coefficients for which
the fundamental solutions are transition density functions of Markov processes
in the four dimensional vector space over the field of p-adic numbers.
1. Introduction
The stochastic processes over the p-adics, or more generally over ultrametric
spaces, have attracted a lot of attention during the last thirty years due to their
connections with models of complex systems, such as glasses and proteins, see
e.g. [2], [3], [4], [6], [10], [13], [14], [15], [23], [22], and the references therein. In
particular, the Markov stochastic process over the p-adics whose transition density
functions are fundamental solutions of pseudodifferential equations of parabolictype have appeared in several new models of complex systems. In [12, Chapters 4,
5] A. N. Kochubei presented a general theory for one-dimensional parabolic-type
pseudodifferential equations with variable coefficients, whose fundamental solutions
are transition density functions for Markov processes in the p-adic line. Taking into
account the physical motivations before mentioned, it is natural to try to develop
a general n-dimensional theory for pseudodifferential equations of parabolic-type
over the p-adics. In this article using the results of [5]-[12, Chapter 4] we present
four-dimensional analogs of the pseudodifferential equations of parabolic-type with
variable coefficients studied by A. N. Kochubei in [12], see also [11]. We should
mention that other types of n-dimensional pseudodifferential equations have bee
studied in [23], [24], [18], [9], [21], [6], among others. To explain the novelty of our
contribution, we begin by saying that the theory developed in [12] depends heavily
on an explicit formula for the Fourier transform of the Riesz kernel associated
to the symbol of the Vladimirov pseudodifferential operator. It turns out, that
this formula is a particular case of the functional equation satisfied for certain
local zeta functions (i.e. distributions in an arithmetical framework) attached to
quadratic forms. This functional equation was established by S. Rallis and G.
Schiffmann in [17], see also the references cited in [5]. In [5] the authors studied
the Riesz kernels and pseudodifferential operators associated to elliptic quadratic
forms of dimensions two and four. Among several results, we determined explicit
formulas for the Fourier transform of the Riesz kernels. In this article we use these
2000 Mathematics Subject Classification. Primary 35S99, 47S10, 60J25; Secondary 11S85.
Key words and phrases. Pseudodifferential operators, Markov processes, non-Archimedean
analysis, p-adic fields.
The second author was partially supported by CONACYT under Grant # 127794.
1
2
O. F. CASAS-SÁNCHEZ AND W. A. ZÚÑIGA-GALINDO
results jointly with classical techniques for parabolic equations to study the Cauchy
problem for a new class of parabolic-type pseudodifferential equations for which the
fundamental solutions are transition density functions of Markov processes in the
four dimensional vector space over the field of p-adic numbers.
The article is organized as follows. In Section 2, we fix the notation and collect
some results about Riesz kernels associated to elliptic quadratic forms of dimension
four. In Section 3 we introduce the heat kernels and certain pseudodifferential operators attached to elliptic quadratic forms and show some basic results needed for
the other sections. In Section 5 we study the Cauchy problem for pseudodifferential
equations with constant coefficients, see Theorem 5.1. In Section 6 we study the
existence and uniqueness of the Cauchy problem for pseudodifferential equations
with variable coefficients, see Theorems 6.3, 6.4, we also discuss the probabilistic
meaning of the fundamental solutions, see Theorem 6.5.
2. Preliminaries
In this section we fix the notation and collect some basic results on p-adic analysis
that we will use through the article. For a detailed exposition on p-adic analysis
the reader may consult [1], [20], [22].
2.1. The field of p-adic numbers. Along this article p will denote a prime number different from 2. The field of p−adic numbers Qp is defined as the completion
of the field of rational numbers Q with respect to the p−adic norm | · |p , which is
defined as
(
0
if x = 0
a
|x|p =
−γ
p
if x = pγ ,
b
where a and b are integers coprime with p. The integer γ := ord(x), with ord(0) :=
+∞, is called the p−adic order of x. We extend the p−adic norm to Qnp by taking
||x||p := max |xi |p ,
1≤i≤n
for x = (x1 , . . . , xn ) ∈ Qnp .
We define ord(x) = min1≤i≤n {ord(xi )}, then ||x||p = p−ord(x) . Any p−adic number
P∞
x 6= 0 has a unique expansion x = pord(x) j=0 xj pj , where xj ∈ {0, 1, 2, . . . , p − 1}
and x0 6= 0. By using this expansion, we define the fractional part of x ∈ Qp ,
denoted {x}p , as the rational number
(
0
if x = 0 or ord(x) ≥ 0
{x}p =
P−ord(x)−1
ord(x)
j
p
xj p if ord(x) < 0.
j=0
For γ ∈ Z, denote by Bγn (a) = {x ∈ Qnp : ||x − a||p ≤ pγ } the ball of radius
pγ with center at a = (a1 , . . . , an ) ∈ Qnp , and take Bγn (0) := Bγn . Note that
Bγn (a) = Bγ (a1 ) × · · · × Bγ (an ), where Bγ (ai ) := {x ∈ Qp : |x − ai |p ≤ pγ } is the
one-dimensional ball of radius pγ with center at ai ∈ Qp . The ball B0n (0) is equals
the product of n copies of B0 (0) := Zp , the ring of p−adic integers. We denote by
Ω(kxkp ) the characteristic function of B0n (0). For more general sets, say Borel sets,
we use 1A (x) to denote the characteristic function of A.
PARABOLIC-TYPE EQUATIONS WITH VARIABLE COEFFICIENTS
3
2.2. The Bruhat-Schwartz space and Fourier transform. A complex-valued
function ϕ defined on Qnp is called locally constant if for any x ∈ Qnp there exist an
integer l(x) such that
n
.
ϕ(x + x′ ) = ϕ(x) for x′ ∈ Bl(x)
(2.1)
A function ϕ : Qnp → C is called a Bruhat-Schwartz function (or a test function) if it
is locally constant with compact support. The C-vector space of Bruhat-Schwartz
functions is denoted by S(Qnp ) := S. For ϕ ∈ S(Qnp ), the largest of such number
l = l(ϕ) satisfying (2.1) is called the exponent of local constancy of ϕ.
Let S ′ (Qnp ) := S ′ denote the set of all functionals (distributions) on S(Qnp ). All
functionals on S(Qnp ) are continuous.
Set χ(y) = exp(2πi{y}p) for y ∈ Qp . The map χ(·) is an additive character
on Qp , i.e. a continuos map from Qp into the unit circle satisfying χ(y0 + y1 ) =
χ(y0 )χ(y1 ), y0 , y1 ∈ Qp .
Pn
Given ξ = (ξ1 , . . . , ξn ) and x = (x1 , . . . , xn ) ∈ Qnp , we set ξ · x := j=1 ξj xj .
The Fourier transform of ϕ ∈ S(Qnp ) is defined as
Z
(F ϕ)(ξ) =
χ(−ξ · x)ϕ(x)dn x for ξ ∈ Qnp ,
Qn
p
where dn x is the Haar measure on Qnp normalized by the condition vol(B0n ) = 1.
The Fourier transform is a linear isomorphism from S(Qnp ) onto itself satisfying
(F (F ϕ))(ξ) = ϕ(−ξ). We will also use the notation Fx→ξ ϕ and ϕ
b for the Fourier
transform of ϕ.
The Fourier transform F [f ] of a distribution f ∈ S ′ Qnp is defined by
(F [f ] , ϕ) = (f, F [ϕ]) for all ϕ ∈ S Qnp .
The Fourier transform f → F [f ] is a linear isomorphism from S ′ Qnp onto S ′ Qnp .
Furthermore, f = F [F [f ] (−ξ)].
2.3. Elliptic Quadratic Forms and Riesz Kernels. We set
f (x) = x21 − ax22 − px23 + apx24 , f ◦ (ξ) = apξ12 − pξ22 − aξ32 + ξ42
with a ∈ Z a quadratic non-residue
The form f (x) is elliptic, i.e. f (x) =
module p. ξ1 ξ2 ξ3 ξ4
◦
0 ⇔ x = 0, and f (ξ) = apf 1 , −a , −p , ap , hence f ◦ (ξ) is also elliptic. The
following estimate will be used frequently along the article: there exist positive
constants A, B such that
B kxk2p ≤ |f (x) |p ≤ A kxk2p for any x ∈ Qnp ,
(2.2)
c.f. [24, Lemma 1].
The Riesz kernels attached to f and f ◦ satisfy the following functional equation:
Z
Z
1 − ps−2
4
4
|f ◦ (ξ) |−s
|f (z)|s−2 ϕ(z)d
b
z=
(2.3)
p ϕ(ξ)d ξ,
1 − p−s
Q4p \{0}
Q4p \{0}
for ϕ ∈ S Q4p and s ∈ C, c.f. [5, Proposition 2.8]. For further details about Riesz
kernels attached to quadratic forms we refer the reader to [5].
4
O. F. CASAS-SÁNCHEZ AND W. A. ZÚÑIGA-GALINDO
Lemma 2.1. If α > 0, then
|f ◦ (x)|α
p =
1 − pα
1 − p−α−2
Z
|f (ξ)|−α−2
[χ(ξ · x) − 1]d4 ξ.
p
Q4p
Proof. The formula follows from (2.3) by using the argument given in [12] for Proposition 2.3.
3. Heat Kernels
We define the heat kernel attached to f ◦ as
Z
◦
α
◦
Z(x, t) := Z(x, t; f , α, κ) = χ(ξ · x)e−κt|f (ξ)|p d4 ξ
Q4p
for x ∈ Q4p , t > 0, α > 0, and κ > 0. When considering Z(x, t) as a function of
x for t fixed we will write Zt (x). The heat kernels considered in this article are a
particular case of those studied in [24].
Given M ∈ Z, we set
Z
◦
α
(M)
Zt (x) := Ω(p−M ||ξ||p )χ(ξ · x)e−κt|f (ξ)|p d4 ξ
Q4p
for x ∈ Q4p , t > 0, α > 0, and κ > 0. Taking into account that
Ω(p−M ||ξ||p )χ(ξ · x)e−κt|f
◦
(ξ)|α
p
≤ e−κt|f
◦
(ξ)|α
p
≤ e−κtB
α
||ξ||2α
p
∈ L1 (Q4p ),
c.f. (2.2), the Dominated Convergence Theorem implies that
(M)
lim Zt
M→∞
(x) = Zt (x) for x ∈ Q4p and for t > 0.
Proposition 3.1. For x, ξ ∈ Q4p \ {0} the following assertions hold:
(i) |f (x + ξ)|p = |f (x)|p , for
||ξ||p < p−1 ||x||p ;
P∞ (−1)m 1−pαm m m
−αm−2
(ii) Z(x, t) = m=1 m!
;
1−p−αm−2 κ t |f (x)|p
(iii) Z(x + ξ, t) = Z(x, t), for ||ξ||p < p−1 ||x||p ;
(iv) Z(x, t) ≥ 0, for x ∈ Q4p and t > 0.
Proof. (i) Set x = pord(x) u, ξ = pord(ξ) v with ||u||p = ||v||p = 1. Then
|f (x + ξ)|p = |f (pord(x)u + pord(ξ) v)|p = |p2ord(x)f (u + pord(ξ)−ord(x)v)|p
= p−2ord(x)|f (u) + pord(ξ)−ord(x)A|p
for some A ∈ Zp . Note that |f (u)|p ∈ {1, p−1 }, then for ord(ξ) − ord(x) > 1,
|f (u) + pord(ξ)−ord(x)A|p = |f (u)|p and
|f (x + ξ)|p = p−2ord(x)|f (u)|p = |p2ord(x)f (u)|p = |f (x)|p .
(M)
(ii) By using the Taylor expansion of ex and Fubini’s Theorem, Zt (x) can be
rewritten as
Z
∞
X
(−1)m m m
(M)
4
Ω(p−M ||ξ||p )χ(ξ · x)|f ◦ (ξ)|mα
κ t
Zt (x) =
p d ξ.
m!
m=0
Q4p
PARABOLIC-TYPE EQUATIONS WITH VARIABLE COEFFICIENTS
5
By using (2.3) with m 6= 0, we have
Z
4
IM :=
Ω(p−M ||ξ||p )χ(ξ · x)|f ◦ (ξ)|mα
p d ξ
Q4p
1 − pαm
=
1 − p−αm−2
Z
Q4p
αm
Fξ→z Ω(p−M ||ξ||p )χ(ξ · x) |f (z)|p−mα−2 d4 z
1−p
p4M
=
1 − p−αm−2
Z
Ω(pM ||x − z||p )|f (z)|p−mα−2 d4 z.
Q4p
We now use the fact that x 6= 0 is fixed, and take M > 1 + ord(x), changing
variables as z = x − pM y in IM , and using (i), we have
Z
1 − pαm
IM =
Ω(||y||p )|f (x − pM y)|p−mα−2 d4 y
1 − p−αm−2
Q4p
=
1 − pαm
|f (x)|p−mα−2
1 − p−αm−2
Z
Ω(||y||p )d4 y =
1 − pαm
|f (x)|p−mα−2 .
1 − p−αm−2
Q4p
On the other hand, for m = 0,
Z
lim
Ω(p−M ||ξ||p )χ(ξ · x)d4 ξ = 0.
M→∞
Q4p
Therefore
Z(x, t) =
∞
X
1 − pαm
(−1)m m m
|f (x)|p−mα−2 for x ∈ Q4p \ {0}.
κ t
−αm−2
m!
1
−
p
m=1
(iii) It is consequence of (ii) and (i).
(iv) See [24, Theorem 2].
Proposition 3.2. The following assertions hold for any x ∈ Q4p , t > 0:
(i) there exists a positive constant C1 such that Z(x, t) ≤ C1 t(t1/2α + ||x||p )−2α−4 ;
R
∂Z(x, t)
4
−κt|f ◦ (η)|α
= −κ Q4 |f ◦ (η)|α
χ(x · η)d η;
pe
p
∂t
(iii) there exists a positive constant C2 such that
−2α−4
∂Z(x, t)
.
≤ C2 t1/2α + ||x||p
∂t
(ii) Z(·, t) ∈ C 1 ((0, ∞) , R) and
Proof. (i) We first consider the case in which t||x||−2α
≤ 1, then by Proposition 3.1
p
(ii)-(iv) and by (2.2),
∞
∞
X
X
C0m
(C0 B −α )m
m
−2
−4
(t|f (x)|−α
)
≤
B
||x||
(t||x||−2α
)m
p
p
p
m!
m!
m=1
m=1
−2
−4
C0 B −α t||x||−2α
−2α−4
p
= B ||x||p e
− 1 ≤ Ct||x||p
.
Z(x, t) ≤ |f (x)|−2
p
6
O. F. CASAS-SÁNCHEZ AND W. A. ZÚÑIGA-GALINDO
We now consider the case in which t > 0. Take k to be an integer satisfying
1
pk−1 ≤ t 2α ≤ pk . Then by Proposition 3.1 (iv), and by (2.2),
|Z(x, t)| = Z(x, t) ≤
≤
Z
Z
e
−κt|f ◦ (ξ)|α
p 4
d ξ≤
e−C0 t||ξ||p d4 ξ
Z
e−C0 ||η|| d4 η ≤ Ct−2/α .
2α
Q4p
Q4p
e
Z
−C0 kp1−k ξ k
2α
p
4
d ξ≤p
−4k−k
2α
Q4p
Q4p
By combining the above inequalities, see for instance the end of the proof of Proposition 4.5, we get the announced result.
∂Z(x, t)
(ii) The formula for
is obtained by a straightforward calculation. The
∂t
∂Z(x, t)
continuity of Z(·, t) is obtained from the formula for
by using the Domi∂t
nated Convergence Theorem. (iii) This part is proved in the same way as (i). The first part of Proposition 3.2 is a particular case of Theorem 1 in [24]. We
include this proof here due to two reasons: first, it shows a very deep connection
between the functional equation (2.3) and the heat kernels; second, we use this
technique for bounding several types of oscillatory integrals in this article.
Corollary 3.3. (i)
1 ≤ ρ ≤ ∞.
R
Q4p
Zt (x)d4 x = 1 for t > 0; (ii) Zt (x) ∈ Lρ for t > 0 and for
4. Some Results on Operators of type f (∂, α)
4.1. The space Mλ . We denote by Mλ , λ ≥ 0, the C-vector space of locally
constant functions ϕ(x) on Q4p such that |ϕ(x)| ≤ C(1 + ||x||λ ), where C is a
positive constant. If the function ϕ depends also on a parameter t, we shall say
that ϕ ∈ Mλ uniformly with respect to t, if its constant C and its exponent of local
constancy do not depend on t.
Lemma 4.1. If ϕ ∈ M2λ , with 0 ≤ λ < α and α > 0, then
lim+
t→0
Z
Q4p
Z(x − ξ, t)ϕ(ξ)d4 ξ = ϕ(x).
PARABOLIC-TYPE EQUATIONS WITH VARIABLE COEFFICIENTS
7
Proof. By Corollary 3.3 (i) and Proposition 3.2 (i), and the fact that ϕ is locally
constant,
Z
I :=
4
Z(x − ξ, t)ϕ(ξ)d ξ − ϕ(x) =
Z
Z(x − ξ, t)[ϕ(ξ) − ϕ(x)]d4 ξ
Q4p
Q4p
Z
=
Z
≤ C1 t
Z(x − ξ, t)[ϕ(ξ) − ϕ(x)]d4 ξ
kx−ξkp ≥pL
(t1/2α + ||x − ξ||p )−2α−4 |ϕ(ξ) − ϕ(x)| d4 ξ
kx−ξkp ≥pL
= C1 t
Z
(t1/2α + ||z||p )−2α−4 |ϕ(x − z) − ϕ(x)| d4 z.
kzkp ≥pL
By applying the triangle inequality in the last integral and noticing that
Z
Z
1/2α
−2α−4 4
||z||−2α−4
d4 ξ ≤ C0 |ϕ(x)| ,
(t
+ ||z||p )
d ξ ≤ |ϕ(x)|
|ϕ(x)|
p
kzkp ≥pL
kzkp ≥pL
and
Z
4
(t1/2α + ||z||p )−2α−4 ||z||2λ
p d ξ ≤
Z
||z||p −2α+2λ−4 d4 z < ∞,
kzkp ≥pL
kzkp ≥pL
we have
lim I ≤ (C1 + C2 |ϕ(x)|) lim+ t = 0.
t→0+
t→0
4.2. The Operator f (∂, α). Given α > 0, we define the pseudodifferential operator with symbol |f ◦ (ξ)|α
p by
S Q4p
→ C Q4p ∩ L2 Q4p
α
−1
(f (∂, α) ϕ) (x) = Fξ→x
|f ◦ (ξ)|p Fx→ξ ϕ .
α
This operator is well-defined since |f ◦ (ξ)|p Fx→ξ ϕ ∈ L1 Q4p ∩ L2 Q4p . By [5,
Proposition 3.4 (iv)],
Z
ϕ(x − y) − ϕ(x) 4
1 − pα
d y,
(4.1)
(f (∂, α) ϕ) (x) =
−α−2
1−p
|f (y)|α+2
p
Q4p
for ϕ ∈ S Q4p . The operator f (∂, α) can be extended to any locally constant
functions ϕ (x) satisfying
Z
|ϕ(x)| 4
d x < ∞ for some m ∈ Z,
(4.2)
|f (x)|α+2
p
→
ϕ
kxkp ≥pm
c.f. [5, Lemma 4.1].
8
O. F. CASAS-SÁNCHEZ AND W. A. ZÚÑIGA-GALINDO
Note that
(M)
Zt (x)
=
Z
||η||p
χ(x · η)e−κt|f
◦
4
(η)|α
p
d η, with M ∈ N
≤pM
is a locally constant and bounded function, c.f. Proposition 3.2. Furthermore, by
(M)
Proposition 3.2 and (2.2), Zt (x) satisfies condition (4.2), for t > 0.
Lemma 4.2.
(M)
(f (∂, γ)Zt )(x)
(4.3)
Z
=
χ(x · η)|f ◦ (η)|γp e−at|f
◦
(η)|α
p 4
d η,
||η||p ≤pM
for M ∈ N and for t > 0.
(M)
(M)
Proof. Note that if ||ξ||p ≤ p−M , then Zt (x − ξ) = Zt (x). In addition, since
(M)
(M)
Zt (x) satisfies condition (4.2), we can use formula (4.1) to compute (f (∂, γ)Zt )(x)
as follows:
Z
h
i
1 − pγ
(M)
(M)
(M)
−γ−2
(f (∂, γ)Zt )(x) =
|f
(ξ)|
Z
(x
−
ξ)
−
Z
(x)
d4 ξ
p
t
t
1 − p−γ−2
=
Q4p
Z
γ
1−p
1 − p−γ−2
||ξ||p >p−M
Z
γ
+
1−p
1 − p−γ−2
||ξ||p ≤p−M
Z
γ
=
1−p
1 − p−γ−2
=
||ξ||p >p−M
Z
γ
1−p
1 − p−γ−2
h
i
(M)
(M)
|f (ξ)|p−γ−2 Zt (x − ξ) − Zt (x) d4 ξ
h
i
(M)
(M)
|f (ξ)|p−γ−2 Zt (x − ξ) − Zt (x) d4 ξ
h
i
(M)
(M)
|f (ξ)|p−γ−2 Zt (x − ξ) − Zt (x) d4 ξ
|f (ξ)|p−γ−2
Z
=
1−p
1 − p−γ−2
e−at|f
◦
(η)|α
p
=
1−p
1 − p−γ−2
Z
χ(x · η)
||η||p ≤pM
γ
e−at|f
||η||p ≤pM
||ξ||p >p−M
γ
Z
◦
(η)|α
p
χ(x · η)[χ(ξ · η) − 1]d4 ηd4 ξ
|f (ξ)|p−γ−2 [χ(ξ · η) − 1]d4 ξd4 η
||ξ||p >p−M
Z
e−at|f
||η||p ≤pM
Z
=
◦
(η)|α
p
χ(x · η)


Z


|f (ξ)|p−γ−2 [χ(ξ · η) − 1]d4 ξ
Q4p
|f ◦ (η)|γp e−at|f
◦
(η)|α
p





d4 η
χ(x · η)d4 η, c.f. Lemma 2.1.
||η||p ≤pM
By Proposition 3.1 (iii) and Proposition 3.2 (i), (f (∂, γ)Zt )(x) is well-defined for
x 6= 0 and for 0 < γ ≤ α.
Proposition 4.3.
(4.4)
(f (∂, γ)Zt )(x) =
Z
Q4p
|f ◦ (η)|γp e−κt|f
◦
(η)|α
χ(x · η)d4 η, for 0 < γ ≤ α, t > 0.
PARABOLIC-TYPE EQUATIONS WITH VARIABLE COEFFICIENTS
9
◦
α
Proof. By (2.2), |f ◦ (·)|γp e−κt|f (·)|p ∈ L1 Q4p for t > 0, then from (4.3), by the
Dominated Convergence Theorem, we obtain
Z
◦
α
(4.5)
lim (f (∂, γ)Z (M) )(x, t) = χ(x · η)|f ◦ (η)|γp e−κt|f (η)|p d4 η, for t > 0.
M→∞
Q4p
On the other hand, fixing an x 6= 0, by Proposition 3.1 (i),
Z
h
i
1 − pγ
(M)
(M)
(M)
−γ−2
(f (∂, γ)Zt )(x) =
Z
(x
−
ξ)
−
Z
(x)
d4 ξ.
|f
(ξ)|
t
t
p
1 − p−γ−2
||ξ||p >p−1 ||x||p
Finally, by the Dominated Convergence Theorem and (4.5), we have
Z
◦
α
(M)
lim (f (∂, γ)Zt )(x) = (f (∂, γ)Z)(x, t) = χ(x · η)|f ◦ (η)|γp e−κt|f (η)|p d4 η.
M→∞
Q4p
Finally, we note that the right-hand side of (4.4) is continuous at x = 0.
Corollary 4.4.
∂Z(x,t)
∂t
= −κ(f (∂, α)Z)(x, t) for t > 0.
Proof. The formula follows from Propositions 4.3 and 3.2 (ii).
Proposition 4.5. If 0 < γ ≤ α, then
|(f (∂, γ)Zt )(x)| ≤ C(t1/2α + ||x||p )−2γ−4 , for x ∈ Q4p and for t > 0.
Proof. By reasoning as in the proof of Proposition 3.1 (ii), we have
∞
X
(−1)m m m
1 − pαm+γ
(4.6)
(f (∂, γ)Zt )(x) =
|f (x)|p−mα−γ−2 .
κ t
−αm−γ−2
m!
1
−
p
m=1
If t||x||−2α ≤ 1, from (4.6) and (2.2), we obtain
(4.7)
∞
X
Cm
m
−2γ−4
(t|f (x)|−α
.
p ) ≤ C1 ||x||p
m!
m=1
|(f (∂, γ)Zt )(x)| ≤ |f (x)|p−γ−2
On other hand, take k such that pk−1 ≤ t1/2α ≤ pk . From (4.4) by using (2.2), we
get
Z
Z
−aB α ||p−(k−1) η||2α
−κtB α ||η||2α
p dη
p dη ≤ Aγ
||η||2γ
e
|(f (∂, γ)Zt )(x)| ≤ Aγ ||η||2γ
p e
p
Q4p
(4.8)
= Aγ p−4(k−1)−2γ(k−1)
Z
Q4p
−aB
||ξ||2γ
p e
α
||ξ||2α
4
p
d ξ ≤ Ct−4−2γ/2α .
Q4p
The announced results follows from inequalities (4.7)-(4.8). Indeed, t||x||−2α
≤1
p
implies that ||x||p ≥
||x||p
2
+
t1/2α
2 ,
||x||−2γ−4
p
and hence
−2γ−4
.
≤ 22γ+4 ||x||p + t1/2α
Now, if t||x||−2α > 1, then t1/2α >
t−4−2γ/2α
t1/2α
2
||x||
+ 2 p and
−2γ−4
.
< 22γ+4 t1/2α + ||x||p
10
O. F. CASAS-SÁNCHEZ AND W. A. ZÚÑIGA-GALINDO
Corollary 4.6.
Z
(f (∂, γ)Zt )(x)d4 x = 0 for t > 0.
Q4p
5. The Cauchy problem
Along this section, we fix the domain (Dom (f )) of the operator f (∂, α) to be the
C-vector space of locally constant functions satisfying (4.2), and f (∂, α)ϕ is given
by (4.1) for ϕ ∈ Dom (f ). Note that M2λ ⊂ Dom (f ) for λ < α.
In this section we study the following Cauchy problem:
(5.1)



∂u(x,t)
∂t
+ κf (∂, γ)u(x, t) = g(x, t),
x ∈ Q4p ,
0 < t ≤ T,
u(x, 0) = ϕ(x)
where κ > 0, α > 0, T > 0, ϕ ∈ M2λ , g (x, t) ∈ M2λ uniformly in t, 0 ≤
λ < α, g(x, t) is continuous in (x, t), and u : Q4p × [0, T ] → C is an unknown
function. We say that u (x, t) is a solution of (5.1), if u (x, t) is continuous in (x, t),
u (·, t) ∈ Dom (f ) for t ∈ [0, T ], u (x, ·) is continuously differentiable for t ∈ (0, T ],
u (x, t) ∈ M2λ uniformly in t, and u satisfies (5.1) for all t > 0.
Theorem 5.1. The function
u(x, t) =
Z
Z(x − y, t)ϕ(y)d4 y +
Zt
0
Q4p
is a solution of Cauchy problem (5.1).


Z

4 
 Z(x − y, t − θ)g(y, θ)d y  dθ
Q4p
The proof of the theorem will be accomplished through the following lemmas.
Lemma 5.2. Assume that g ∈ M2λ , 0 ≤ λ < α, uniformly with respect to θ. Then
the function


Zt Z


u2 (x, t, τ ) :=  Z(x − y, t − θ)g(y, θ)d4 y  dθ
τ
Q4p
belongs to M2λ uniformly with respect to t and τ .
Proof. We first note that u2 (x, t, τ ) has the same exponent of local constancy as
g, and thus it does not depend on t and τ . We now show that |u2 (x, t, τ )| ≤
C0 (1 + ||x||2λ
p ). By Proposition 3.2 (i),


Zt Z


|u2 (x, t, τ )| ≤  |Z(x − y, t − θ)||g(y, θ)|d4 y  dθ
τ
≤ C1
Q4p
Zt
τ


(t − θ) 
Z
Q4p


((t − θ)1/2α + ||x − y||p )−2α−4 (1 + ||y||2λ
p )dy  dθ.
Now the result follows from the following estimation.
PARABOLIC-TYPE EQUATIONS WITH VARIABLE COEFFICIENTS
11
Assertion([18, Proposition 2]). If b > 0, 0 ≤ λ < 2α, and x ∈ Q4p , then
Z
−2α−4
4
−2α
,
1 + ||x||2λ
(b + ||x − ξ||p )
||ξ||2λ
(5.2)
p
p d ξ ≤ Cb
Q4p
where the constant C does not depend on b or x.
Lemma 5.3. Assume that g ∈ M2λ , 0 ≤ λ < α, uniformly with respect to θ. Then


Zt Z
∂u2 (x, t, τ )
 ∂Z(x − ξ, t − θ)

= g(x, t) + 
[g(ξ, θ) − g(x, θ)]d4 ξ  dθ.
∂t
∂t
τ
Q4p
Proof. Set
t−h Z
Z
u2,h (x, t, τ ) :=
dθ Z(x − ξ, t − θ)g(ξ, θ)d4 ξ,
τ
Q4p
where h is a small positive number. By differentiating uh under the sign of integral
t−h Z
Z
Z
∂Z(x − ξ, t − θ)
∂u2,h
4
=
g(ξ, θ)d ξ + Z(x − ξ, h)g(ξ, t − h)d4 ξ
dθ
∂t
∂t
τ
Q4p
Q4p
t−h Z
Z
∂Z(x − ξ, t − θ)
=
dθ
[g(ξ, θ) − g(x, θ)]d4 ξ
∂t
τ
Q4p
t−h
Z
Z
Z
∂Z(x − ξ, t − θ) 4
d ξ + Z(x − ξ, h)[g(ξ, t − h) − g(ξ, t)]d4 ξ
+
g(x, θ)dθ
∂t
τ
+
Z
Q4p
Q4p
Z(x − ξ, h)g(ξ, t)d4 ξ.
Q4p
The first integral contains no singularity at t = θ due to Proposition 3.2 (iii) and
the local constancy of g. By Proposition 3.2 (i) and Corollary 3.3 (i), the second
integral is equal to zero. The third integral can be written as the sum of the integrals
over {ξ ∈ Q4p | ||x − ξ||p ≤ pM }, where M is the exponent of local constancy of g,
and the complement of this set. The first integral tends to zero when h → 0+ due
to the uniform local constancy of g, while the other tends to zero when h → 0+
due to Proposition 3.2 (i) and condition λ < α. Finally, the fourth integral tends
to g(x, t) as h → 0+ , c.f. Lemma 4.1.
For ϕ ∈ M2λ , 0 ≤ λ < α, we set
Z
u1 (x, t) := Z(x − y, t)ϕ(y)d4 y for t > 0.
Q4p
Lemma 5.4. Assume that ϕ ∈ M2λ , 0 ≤ λ < α, then the following assertions
hold:
(i) u1 (x, t) belongs
M2λ uniformly with respect to t;
R to
∂Z
4
1
(x,
t)
=
(ii) ∂u
∂t
Q4 ∂t (x − y, t)ϕ(y)d y for t > 0.
p
12
O. F. CASAS-SÁNCHEZ AND W. A. ZÚÑIGA-GALINDO
Proof. (i) The proof is similar to that of Lemma 5.2.
(ii) By Proposition 3.2 (ii),
Z u1 (x, t + h) − u1 (x, t)
Z(x − y, t + h) − Z (x − y, t)
lim
ϕ(y)d4 y
= lim
h→0
h→0
h
h
Q4p
= lim
Z
h→0
Q4p
∂Z
(x − y, τ )ϕ(y)d4 y
∂t
where τ is between t and t + h. Now the result follows from Proposition 3.2 (iii) by
applying the Dominated Convergence Theorem.
Lemma 5.5. Assume that λ < γ ≤ α. Then
Z
(f (∂, γ)u1 )(x, t) = (f (∂, γ)Zt )(x − y)ϕ(y)d4 y for t > 0.
Q4p
Proof. By Lemma 5.4 (i), u1 (x, t) belongs to the domain of f (∂, γ) for t > 0 and
for λ < γ ≤ α, then for any L ∈ N, the following integral exists:
Z
1 − pγ
|f (y)|p−γ−2 [u1 (x − y, t) − u1 (x, t)] d4 y
1 − p−γ−2
||y||p >p−L
1 − pγ
=
1 − p−γ−2
Z
|f (y)|p−γ−2
||y||p >p−L
"Z
#
[Zt (x − y − ξ) − Zt (x − ξ)] ϕ(ξ)d4 ξ d4 y.
Q4p
By using Fubini’s Theorem, see (2.2), Proposition 3.2 (i),


Z
Z
γ

 1−p
|f (y)|p−γ−2 [Zt (x − y − ξ) − Zt (x − ξ)] d4 y  ϕ(ξ)d4 ξ

−γ−2
1−p
Q4p
(5.3) =:
Z
||y||p >p−L
(γ,L)
ϕ(ξ)Zt
(x − ξ)d4 ξ.
Q4p
By fixing a positive integer M , the last integral in (5.3) can be expressed as
Z
Z
(γ,L)
(γ,L)
ϕ(ξ)Zt
(x − ξ)d4 ξ.
ϕ(ξ)Zt
(x − ξ)d4 ξ +
(5.4)
||x−ξ||p ≥p−M
||x−ξ||p <p−M
(γ,L)
note that if ||x||p ≥ p−M and M < L − 1, then, by Proposition 3.1 (iii), Zt
(f (∂, γ)Zt )(x), and
Z
1 − pγ
lim
|f (y)|p−γ−2 [u1 (x − y, t) − u1 (x, t)] d4 y
L→∞ 1 − p−γ−2
=
Z
||x−ξ||p≥p−M
||y||p >p−L
ϕ(ξ)(f (∂, γ)Zt )(x − ξ)d4 ξ + lim
Z
L→∞
||x−ξ||p <p−M
(γ,L)
ϕ(ξ)Zt
=
(x − ξ)d4 ξ,
PARABOLIC-TYPE EQUATIONS WITH VARIABLE COEFFICIENTS
13
for M < L − 1. Now by using twice Fubini’s theorem, see (2.2), Proposition 3.2 (i),
and Proposition 3.1 (iii), we have
Z
lim
(γ,L)
L→∞
||x−ξ||p <p−M

γ
 1−p

1 − p−γ−2
ϕ(ξ)Zt
Z
||y||p >p−L
Z
= lim

L→∞
||y||p >p−L
 1−p

1 − p−γ−2
=
Q4p
=
Z


ϕ(ξ) [Zt (x − y − ξ) − Zt (x − ξ)] d4 ξ  d4 y
Z
γ
 1−p
|f (y)|p−γ−2 
1 − p−γ−2
Z
ϕ(ξ)×


||x−ξ||p <p−M
=
L→∞
||x−ξ||p <p−M
|f (y)|p−γ−2 ×
||x−ξ||p <p−M
Z
Z

|f (y)|p−γ−2 [Zt (x − y − ξ) − Zt (x − ξ)] d4 y  d4 ξ
Z
γ
(x − ξ)d4 ξ = lim
||x−ξ||p <p−M

 1 − pγ
ϕ(ξ) 
1 − p−γ−2
Z


ϕ(ξ) [Zt (x − y − ξ) − Zt (x − ξ)] d4 ξ  d4 y


|f (y)|p−γ−2 [Zt (x − y − ξ) − Zt (x − ξ)] d4 y  d4 ξ
Q4p
(f (∂, γ)Zt )(x − ξ)ϕ(ξ)d4 ξ.
||x−ξ||p
<p−M
Lemma 5.6. If λ < γ ≤ α, then
(f (∂, γ)u2 )(x, t, τ ) =
Zt
τ


Z


 (f (∂, γ)Z)(x − y, t − θ)g(y, θ)d4 y  dθ for t > 0.
Q4p
Proof. Let


t−h Z
Z

4 
u2,h (x, t, τ ) =:
 Z(x − y, t − θ)g(y, θ)d y  dθ
τ
Q4p
where h is a small positive number such that 0 < h < t − τ . Set
Z (γ,L)(x, t) =
1 − pγ
1 − p−γ−2
Z
||y||p >p−L
|f (y)|p−γ−2 [Z(x − y, t) − Z(x, t)] d4 y.
14
O. F. CASAS-SÁNCHEZ AND W. A. ZÚÑIGA-GALINDO
By the Fubini Theorem
Z
1 − pγ
1 − p−γ−2
|f (y)|p−γ−2 [u2,h (x − y, t, τ ) − u2,h (x, t, τ )] d4 y
||y||p >p−L
(5.5)
t−hZ
Z
=
Z (γ,L)(x − ξ, t − θ)g(ξ, θ)d4 ξdθ.
τ Q4p
Note that
Z (γ,L) (x, t) =
Z
1 − pγ
1 − p−γ−2
||y||p


Z
◦
α


−γ−2
χ(ξ · x) [χ(−ξ · y) − 1] e−κt|f (ξ)|p d4 ξ  d4 y
 |f (y)|p
Q4p
=
Z
×
>p−L
◦
χ(ξ · x)e−κt|f
(ξ)|α
p
Q4p

 1 − pγ

1 − p−γ−2
Z
||y||p >p−L


|f (y)|p−γ−2 [χ(−ξ · y) − 1] d4 y  d4 ξ
=
Z
χ(ξ · x)e−at|f
◦
(ξ)|α
p
PL (ξ)d4 ξ,
Q4p
where
Z
1 − pγ
1 − p−γ−2
PL (ξ) =
|f (y)|p−γ−2 [χ(−ξ · y) − 1] d4 y.
||y||p >p−L
On the other hand, by (2.2),
1 − pγ
1 − p−γ−2
|PL (ξ)| ≤ B −γ−2
Z
||y||−2γ−4
|χ(−ξ · y) − 1| d4 y,
p
||y||p >p−L
and by using a similar reasoning to the one used in [12, p. 142], we have
|PL (ξ)| ≤ C||ξ||2γ
p
whence
(5.6)
Z
(γ,L)
(x, t) ≤
Z
e
−κt|f ◦ (ξ)|α
p
4
|PL (ξ)| d ξ ≤ C
Z
e−κt|f
◦
(ξ)|α
p
4
′
||ξ||2γ
p d ≤ C ,
Q4p
Q4p
where C ′ is a positive constant, which not depend on x, t ≥ h + τ, L.
By writing the right-hand side of (5.5) as
(5.7)
t−h
Z
Z
Z (γ,L) (x − ξ, t − θ)g(ξ, θ)d4 ξdθ
τ ||x−ξ||p ≥p−K
+
t−h
Z
Z
τ ||x−ξ||p <p−K
Z (γ,L) (x − ξ, t − θ)g(ξ, θ)d4 ξdθ,
PARABOLIC-TYPE EQUATIONS WITH VARIABLE COEFFICIENTS
15
where K is a fixed natural number. Now the result follows by taking limit L → ∞
in (5.5). Indeed, for the first integral in (5.7), if ||x − ξ||p ≥ p−K and L > K + 1,
then Z (γ,L) (x − ξ, t − θ) = (f (∂, γ)Z)(x − ξ, t − θ). For the second integral in (5.7),
we use (5.6) and the Dominated Convergence Theorem.
5.1. Proof of Theorem 5.1. By Lemmas 5.2 and 5.4 (i), u(x, t) ∈ M2λ uniformly
with respect to t, and by Lemma 4.1 u(x, t) satisfies the initial condition. By
Lemmas 5.3-5.6 and Corollary 4.4, u(x, t) is a solution of Cauchy problem (5.1).
6. Parabolic-type equations with variable coefficients
Fix n + 1 positive real numbers satisfying
0 < α1 < α2 < · · · < αn < α,
with α > 1. We also fix n + 2 functions ak (x, t), k = 0, . . . , n and b(x, t) from
Q4p × [0, T ] to R, here T is a fixed positive constant. We assume that ak (x, t),
k = 0, . . . , n and b(x, t) satisfy the following conditions:
(i) they belong to M0 , with respect to x, uniformly in t ∈ [0, T ];
(ii) they satisfy the Hölder condition in t with exponent ν ∈ (0, 1] uniformly in x.
In addition we assume the uniform parabolicity condition a0 (x, t) ≥ µ > 0.
We set
n
X
F (∂, α1 , α2 , · · · , αn ) :=
ak (x, t)f (∂, αk ) + b(x, t)I.
k=1
Note that F (∂, α1 , α2 , · · · , αn ) M2λ ⊂ M2λ for λ < α1 , c.f. (4.1) .
In this section we study the following initial value problem:
(6.1)

∂u(x, t)


+ a0 (x, t)(f (∂, α)u)(x, t) + (F (∂, α1 , α2 , · · · , αn ) u) (x, t) = g(x, t)

∂t



u(x, 0) = ϕ(x), x ∈ Q4p , t ∈ (0, T ],
where x ∈ Q4p , t ∈ (0, T ], ϕ ∈ M2λ with 0 ≤ λ < α1 (if a1 (x, t) = · · · = an (x, t) ≡ 0,
then we shall assume that 0 ≤ λ < α), and g(x, t) is continuous in (x, t), and
g (x, t) ∈ M2λ uniformly in t ∈ [0, T ], 0 ≤ λ < α1 .
In this section we find the solution of the general problem (6.1). The technique
used here is an adaptation of the classical Archimedean techniques, see e.g. [8],
[16]. In the p-adic setting the technique was introduced by A. N. Kochubei in [11].
Our presentation is highly influenced by Kochubei’s book [12]. The proofs of some
theorems are very similar to the corresponding in [12] for this reason we will omit
them.
The first step of the construction of a fundamental solution is to study the
parametrized fundamental solution Z(x, t, y, θ) for the Cauchy problem:
 ∂u(x,t)

+ a0 (y, θ)(f (∂, α)u)(x, t) = 0
∂t

u (x, 0) = ϕ (x) ,
where y ∈ Q4p and θ > 0 are parameters. By the results of Section 3, we have
Z
◦
α
Z(x, t, y, θ) = χ(ξ · x)e−a0 (y,θ)t|f (ξ)|p d4 ξ,
Q4p
16
O. F. CASAS-SÁNCHEZ AND W. A. ZÚÑIGA-GALINDO
and if x 6= 0, then
∞
X
(−1)m
1 − pαm
Z(x, t, y, θ) =
(a0 (y, θ)t)m |f (x)|p−αm−2 ,
−αm−2
m!
1
−
p
m=1
c.f. Proposition 3.1 (ii). By the Propositions 3.2, 4.3, 4.5, and Corollaries 3.3 (i)
and 4.6, we have
(6.2)
Z(x, t, y, θ) ≤ Ct(t1/2α + ||x||p )−2α−4 ,
x ∈ Q4p ,
t > 0,
(6.3)
|(f (∂, γ)Z)(x, t, y, θ)| ≤ C(t1/2α + ||x||p )−2γ−4 ,
x ∈ Q4p ,
(6.4)
Z
∂Z(x, t, y, θ)
−a0 (y,θ)t|f ◦ (η)|α
χ(x · η)d4 η,
= −a0 (y, θ) |f ◦ (η)|α
pe
∂t
t > 0,
Q4p
(6.5)
(6.6)
−2α−4
∂Z(x, t, y, θ)
,
≤ C t1/2α + ||x||p
∂t
(f (∂, γ)Z)(x, t, y, θ) =
Z
Q4p
|f ◦ (η)|γp e−a0 (y,θ)t|f
Z
(6.7)
◦
(η)|α
χ(x·η)d4 η,
x 6= 0,
0 < γ ≤ α,
Z(x, t, y, θ)d4 x = 1,
Q4p
(6.8)
Z
(f (∂, γ)Z)(x, t, y, θ)d4 x = 0,
Q4p
where the constants do not depend on y, θ.
Lemma 6.1. There exists a positive constant C, such that
(6.9)
Z
∂Z(x − y, t, y, θ) 4
d y ≤ C.
∂t
Q4p
Proof. The proof follows from (6.7), (6.4), (6.5) by using the resoning given in [12]
for Lemma 4.5.
Consider the parametrized heat potential
u(x, t, τ ) :=
Zt Z
Z(x − y, t − θ, y, θ)g(y, θ)d4 ydθ,
τ Q4p
with g ∈ M2λ , 0 ≤ λ < α1 , uniformly with respect to θ, and continuous in (y, θ). By
using (6.2) we obtain as Lemma 5.2 that u(x, t, τ ) is locally constant and belongs
PARABOLIC-TYPE EQUATIONS WITH VARIABLE COEFFICIENTS
17
to M2λ uniformly with respect to t, τ .
Also we have
Zt Z
∂u
∂Z(x − y, t − θ, y, θ)
(x, t, τ ) = g(x, t) +
[g(y, θ) − g(x, θ)] d4 ydθ
∂t
∂t
τ Q4p
(6.10)
+
Zt Z
∂Z(x − y, t − θ, y, θ)
g(x, θ)d4 ydθ,
∂t
τ Q4p
see Lemma 6.1, and
(6.11)
(f (∂, γ)u)(x, t, τ ) =
Zt Z
Z (γ) (x − y, t − θ, y, θ)g(y, θ)d4 ydθ,
τ Q4p
with Z (γ) (x, t, y, θ) := f (∂, γ)Z (x, t, y, θ), for λ < γ < α, and
(f (∂, α)u)(x, t, τ ) =
Zt Z
Z (α) (x − y, t − θ, y, θ) [g(y, θ) − g(x, θ)] d4 ydθ
τ Q4p
(6.12)
+
Zt Z h
τ Q4p
i
Z (α) (x − y, t − θ, y, θ) − Z (α) (x − y, t − θ, x, θ) g(x, θ)d4 ydθ.
As in [12] we look for a fundamental solution of (6.1) of the form
(6.13) Γ(x, t, ξ, τ ) = Z(x − ξ, t − τ, ξ, τ ) +
Zt Z
Z(x − η, t − θ, η, θ)φ(η, θ, ξ, τ )d4 ηdθ.
0 Q4p
By using formally the formulas given above, we can see that φ(x, t, ξ, τ ) is a solution
of the integral equation
(6.14)
φ(x, t, ξ, τ ) = R(x, t, ξ, τ ) +
Zt Z
R(x, t, η, θ)φ(η, θ, ξ, τ )d4 ηdθ
0 Q4p
where
R(x, t, ξ, τ ) = [a0 (ξ, τ ) − a0 (x, t)] Z (α) (x − ξ, t − τ, ξ, τ )
−
n
X
ak (x, t)Z (αk ) (x − ξ, t − τ, ξ, τ ) − b(x, t)Z(x − ξ, t − τ, ξ, τ ).
k=1
Integral equation (6.14) can be solved by the methods of successive approximations:
(6.15)
φ(x, t, ξ, τ ) =
∞
X
Rm (x, t, ξ, τ )
m=1
where
R1 (x, t, ξ, τ ) := R(x, t, ξ, τ )
18
O. F. CASAS-SÁNCHEZ AND W. A. ZÚÑIGA-GALINDO
and
Rm+1 (x, t, ξ, τ ) :=
Zt Z
R(x, t, η, θ)Rm (η, θ, ξ, τ )d4 ηdθ.
0 Q4p
In order to prove the convergence of series (6.15), we need the following lemma
which is an easy variation of the Lemmas 6 and 7 given in [19].
We set αn+1 := α (1 − ν) > αn .
Lemma 6.2. The following estimations hold:
n+1
−2αk −4
X
|R(x, t, ξ, τ )| ≤ C
(t − τ )1/2α + ||x − ξ||p
k=1
and
|Rm (x, t, ξ, τ )| ≤ DM
mΓ
Γ
ν m n+1
X
2α (t
mν
2α
k=1
1/2α
− τ)
+ ||x − ξ||p
−2αk −4
where C, M and D are a positive constants and Γ (·) is the Archimedean Gamma
function.
Lemma 6.2 also implies
(6.16)
|φ(x, t, ξ, τ )| ≤ C
n+1
X
(t − τ )
1/2α
+ ||x − ξ||p
k=1
−2αk −4
.
Theorem 6.3. The function
Zt Z
Z
4
Γ(x, t, ξ, τ )g(ξ, τ )d4 ξdτ
(6.17)
u(x, t) = Γ(x, t, ξ, 0)ϕ(ξ)d ξ +
0 Q4p
Q4p
Qnp
which is continuous on
× [0; T ], continuously differentiable in t ∈ (0, T ], and
belonging to M2λ uniformly with respect to t is a solution of Cauchy problem (6.1).
The fundamental solution Γ(x, t, ξ, τ ), x, ξ ∈ Q4p , 0 ≤ τ < t ≤ T , is of the form
(6.18)
with
(6.19)
Γ(x, t, ξ, τ ) = Z(x − ξ, t − τ, ξ, τ ) + W (x, t, ξ, τ )
i−2α−4
h
|W (x, t, ξ, τ )| ≤ C (t − τ )1+ν (t − τ )1/2α + ||x − ξ||p
+ C (t − τ )
n+1
Xh
1/2α
(t − τ )
+ ||x − ξ||p
k=1
i−2αk −4
.
Furthermore Z(x, t, y, θ) satisfies the estimates (6.2), (6.3), (6.5), (6.9).
Proof. Denote for u1 (x, t) and u2 (x, t) the first and second summands in the righthand side of (6.17). Substituting (6.13) into (6.17) we get and
Z
(6.20)
u1 (x, t) = Z(x − ξ, t, ξ, 0)ϕ(ξ)d4 ξ
Q4p
+
Z tZ
0
Q4p
Z(x − η, t − θ, η, θ)G(η, θ)d4 ηdθ,
PARABOLIC-TYPE EQUATIONS WITH VARIABLE COEFFICIENTS
where
G(η, θ) =
Z
19
φ(η, θ, ξ, 0)ϕ(ξ)d4 ξdτ.
Q4p
and
(6.21)
u2 (x, t) =
Z tZ
Z(x − ξ, t − τ, ξ, τ )g(ξ, τ )d4 ξdτ
Z tZ
Z(x − η, t − θ, η, θ)F (η, θ)d4 ηdθ,
0
+
0
Q4p
Q4p
where
F (η, θ) =
Z
0
θ
Z
φ(η, θ, ξ, τ )g(ξ, τ )d4 ξdτ.
Q4p
Now by (6.16) and (5.2),
|F (η, θ)| ≤ C, and |G(η, θ)| ≤ Cθ−αn+1 /α
for all η ∈ Q4p and θ ∈ (0, T ]. In addition the functions F and G are uniformly
locally constant. Indeed, by the recursive definition of the function φ, we see that
if N is a local constancy exponent for all the functions g, ϕ, ak , b, Z (αk ) and Z,
and if |δ| ≤ p−N , then
φ(η + δ, θ, ξ + δ, τ ) = φ(η, θ, ξ, τ ),
therefore
F (η + δ, θ) = F (η, θ), and G(η + δ, θ) = G(η, θ).
Thus, the potentials in the expressions for u1 (x, t) and u2 (x, t) satisfy the conditions
under which the differentiation formulas (6.10), (6.11), (6.12) were obtained. By
using these formulas one verifies after some simple transformations that u(x, t) is a
solution of the equation in (6.1).
We now show that u(x, t) → ϕ(x) as t → 0+ . Due to (6.20) and (6.21), it is
sufficient to verify that
Z
v(x, t) = Z(x − ξ, t, ξ, 0)ϕ(ξ)d4 ξ → ϕ(x) as t → 0+ .
Q4p
By virtue of equation (6.7),
Z
v(x, t) = [Z(x − ξ, t, ξ, 0) − Z(x − ξ, t, x, 0)] ϕ(ξ)d4 ξ
Q4p
+
Z
Z(x − ξ, t, x, 0) [ϕ(ξ) − ϕ(x)] d4 ξ + ϕ(x).
Q4p
We now use that Z (as a function of the its third argument) and ϕ are locally
constant, then the integrals in the previous formula are performed over the set
ξ ∈ Q4p ; ||x − ξ||p ≥ p−N for some N ∈ N.
By applying (6.2) we see that both integrals tend to zero as t → 0+ .
20
O. F. CASAS-SÁNCHEZ AND W. A. ZÚÑIGA-GALINDO
6.1. The uniqueness of the solution of the Cauchy problem. The technique
used in [12, Theorem 4.5] for establishing the uniqueness of the solution of the
Cauchy problem associated with perturbations of the Vladimirov operator can be
used to show the uniqueness of the Cauchy problem (6.1). Then we state here the
corresponding result without proof.
Theorem 6.4. Assume that the coefficients ak (x, t), k = 0, . . . , n are non-negative,
bounded, continuous functions and that b(x, t) is bounded, continuous function.
Take 0 ≤ λ < α1 (if a1 (x, t) = · · · = an (x, t) ≡ 0, then we shall assume that
0 ≤ λ < α). If u (x, t) is a solution of (6.1) with g (x, t) ≡ 0, such that u ∈ M2λ
uniformly with respect to t, and ϕ (x) ≡ 0, then u (x, t) ≡ 0.
6.2. Probabilistic interpretation. The fundamental solution of (6.1) Γ(x, t, ξ, τ )
is the transition function for a random walk on Q4p . This result can be established
by using classical results on stochastic processes see e.g. [7] and the techniques
given in [12, pp. 161-162]. Formally we have
Theorem 6.5. Assume that the coefficients ak (x, t), k = 0, . . . , n and b(x, t) are
non-negative, bounded, continuous functions. The fundamental solution Γ(x, t, ξ, τ )
is the transition density of a bounded right-continuous Markov process without second kind discontinuities.
6.3. Quadratic forms of dimension two. All the results presented in this article
are valid if for the quadratic forms of type ξ12 − τ ξ22 where τ ∈ Qp r {0} is a not
square of an element of Qp , see [5].
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Universidad Nacional de Colombia, Departamento de Matemáticas, Ciudad Universitaria, Bogotá D.C., Colombia.
E-mail address: ofcasass@unal.edu.co
Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional, Departamento de Matemáticas, Unidad Querétaro, Libramiento Norponiente
#2000, Fracc. Real de Juriquilla. Santiago de Querétaro, Qro. 76230, México.
E-mail address: wazuniga@math.cinvestav.edu.mx